Computing linear response statistics using orthogonal polynomial based estimators: An RKHS formulation
Computing linear response statistics using orthogonal polynomial based estimators: An RKHS formulation
复制标题
使用基于正交多项式的估计器计算线性响应统计数据：RKHS 公式
DOI：
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复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Xiantao Li
中科院分区：
文献类型：
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作者：
He Zhang;J. Harlim;Xiantao Li
研究了利用未扰动动力学时间序列估计外部扰动下的线性响应统计量的问题。这个估计问题的标准方法是采用波动耗散理论，它需要基本未扰动密度的函数形式的知识，这是一般不可用的。为了克服这个问题，我们考虑一个非参数密度估计公式的核嵌入的分布。为了避免与使用径向型核相关的计算开销，我们考虑基于定义在非紧域上的经典正交基构造的“Mercer型”核。虽然所得的表示类似于多项式混沌展开（PCE），通过研究再生核希尔伯特空间（RKHS）的设置，我们建立了一致收敛的估计。更重要的是，RKHS配方允许一个系统地解决一个实际的问题，确定PCE的基础上，通过目标函数的衰减特性，可以使用现有的数据进行量化的一致的估计。在线性响应估计方面，我们的研究不仅为估计量的适定性，而且为响应统计量的适定性提供了实际条件。我们提供了一个理论上的保证，估计的收敛性的基本线性响应统计。最后，我们提供了一个误差界的密度估计，占蒙特-卡罗平均在非独立同分布的时间序列和由于截断的偏差。该误差界有助于理解嵌入Mercer型核的可行性和局限性。在数值上，我们验证了核嵌入线性响应估计的有效性与已知的，但非平凡的平衡密度的两个随机动态。
We study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. A standard approach to this estimation problem is to employ the Fluctuation-Dissipation Theory, which requires the knowledge of the functional form of the underlying unperturbed density that is not available in general. To overcome this issue, we consider a nonparametric density estimator formulated by the kernel embedding of distributions. To avoid the computational expense associated with using radial type kernels, we consider the "Mercer-type" kernels constructed based on the classical orthogonal bases defined on non-compact domains. While the resulting representation is analogous to Polynomial Chaos Expansion(PCE), by studying in the reproducing kernel Hilbert space(RKHS) setting, we establish the uniform convergence of the estimator. More importantly, the RKHS formulation allows one to systematically address a practical question of identifying the PCE basis for a consistent estimation through the decay property of the target functions that can be quantified using the available data. In terms of the linear response estimation, our study provides practical conditions for the well-posedness of not only the estimator but also the well-posedness of the underlying response statistics. We provide a theoretical guarantee for the convergence of the estimator to the underlying linear response statistics. Finally, we offer an error bound for the density estimation that accounts for the Monte-Carlo averaging over non-i.i.d time series and the biases due to truncation. This error bound helps understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernels. Numerically, we verify the effectiveness of the kernel embedding linear response estimator on two stochastic dynamics with known, yet, non-trivial equilibrium densities.
基于分区的贝叶斯多元密度估计方法的收敛率。
DOI：
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发表时间：
2017
期刊：
Advances in neural information processing systems
影响因子：
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作者：
Liu,Linxi;Li,Dangna;Wong,WingHung
通讯作者：
Wong,WingHung
影响因子：
2.3
作者：
Berry, Tyrus;Sauer, Timothy
通讯作者：
Sauer, Timothy